Hellman, Structuralism Without Structures

Structuralism Without Structures

GEOFFREY HELLMAN*

1. Introduction: Approaches to StructuralismAs with many 'isms', 'structuralism' is rooted in some intuitive views ortheses which are capable of being explicated and developed in a varietyof distinct and apparently conflicting ways. One such way, the modal-structuralist approach, was partially articulated in my [1989] (hereinafter'MWON'). That account, however, was incomplete in certain importantrespects bearing on the overall structuralist enterprise. In particular, itwas left open how to treat generally some of the most important structuresor spaces in mathematics, e.g., metric spaces, topological spaces, differ-entiable manifolds, and so forth. This may have left the impression thatsuch structures would have to be conceived as embedded in models of settheory, whose modal-structural interpretation depends on a rather boldconjecture, e.g., the logical possibility of full models of the second-orderZF axioms. Furthermore, the presentation in MWON did not avail itselfof certain technical machinery (developed by Boolos [1985] and Burgess,Hazen, and Lewis [1991]) which can be used to strengthen the programsubstantially. Indeed, these two aspects are closely interrelated; as willemerge, the machinery can be used to fill in the incompleteness so as toavoid dependence on models of set theory. The principal aim of this paperis to take the program forward by elaborating on these developments.

It will be helpful first, however, to remind ourselves of the main intuitiveideas underlying 'structuralism' and to indicate at least roughly where inthe landscape of alternative approaches the one pursued here resides.

One intuitive thesis (one I explicitly highlighted in MWON) is this:Mathematics is the free exploration of structural possibilities, pursued by(more or less) rigorous deductive means.

Vague as this is, it already at least suggests the modern view of geometry,abstract algebra, number systems, and other 'abstract spaces', in which weattempt to characterize the structures of interest by laying down 'axioms'understood as 'defining conditions', which we may be able to show succeed

* Department of Philosophy, University of Minnesota, Minneapolis, MN 55455, U. S. A.hellm001Omaroon.tc.umn.edu

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in their role by producing a proof of their categoricity, and then proceedingto explore their (interesting) consequences. This in turn reveals the impor-tance of second-order logical notions in mathematical foundations, for, asis well known, first-order renditions of defining conditions will inevitablyfail to characterize certain of the most central structures in all of mathe-matics, including the natural-number structure, the reals, the complexes,and initial segments of the cumulative set-theoretic hierarchy. (Cf. Shapiro[1991]; also Mayberry [1994].)

A second intuitive principle, traceable to certain ideas of Dedekind [1888]and widely noted by philosophers and logicians, can be put thus:

In mathematics, it is not particular objects which matter but rather certain'structural' properties and relations, both within and among relevant totalities(domains).

To this one may wish to add:The very identity of individual mathematical objects depends on such struc-tural relations (i.e., on 'relative positions' in structures).

This is illustrated by pointing out that it is nonsensical, for example, to pos-tulate a single real number (as Field [1980], p. 31, entertained; cf. Shapiro[forthcoming]); to be a real number is to be part of a complete, separable,ordered continuum. Particular constructions or definitions (e.g., as conver-gent rational sequences) may, in given contexts, allow one to recover suchstructure, and, by focussing on a particular cdnstruction, it may appearthat one could sensibly potulate a single such item (say, sequence); but wecan only regard this as postulating a real number after we have recoveredthe structure, and so Shapiro's point stands.

Note that, in stating this second intuitive thesis, we have been inten-tionally vague about its scope. Is it understood as saying that all math-ematical reference to objects is to be interpreted structurally (whateverthat means precisely), or does it say, more modestly, that salient casesare? What of notions such as 'finite set' and 'finite sequence' of given ob-jects? In contrast to numbers, the identity of a finite set of objects A, forexample, seems determined by its members without considering its rela-tive position within the naturally associated structure, the totality of finiteA—sets ordered by inclusion, itself a fairly complicated infinitistic object.(Cf. Parsons's [1990] related points concerning 'quasi-concrete' mathemati-cal objects.) And what of mathematical reference to structures themselves?Is there a regress involved in interpreting such reference structurally, and, ifso, is it a vicious one? (Cf. Shapiro [forthcoming].) An adequate structural-ism should somehow account for these apparent differences among mathe-matical concepts. As different approaches may be expected to treat suchmatters differently, let us take the second thesis in its limited, modest sense,allowing for supplementation as a particular view may require.



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So understood, there are at least four main approaches to structuralismthat have been proposed and should be distinguished:

(i) The framework of Model Theory ('MT'), carried out in Set Theory(say, ZFC). Structures are understood as models (sets as domains, togetherwith distinguished relations and possibly individuals), and one can alsospeak of isomorphism types as 'structures' (at least one can if one is carefulin either avoiding or admitting proper classes). (Of course, different choicesof set theory yield different explicanda.) Equivalence of nominally distinctstructures can be defined in terms of 'definitional extension' and relatednotions. (Thus, for example, the full second-order natural-number structurewith just successor distingushed is equivalent to that obtained by addingaddition and multiplication.)

(ii) The framework of Category Theory ('CT') (which itself can be ax-iomatized, as in Mac Lane [1986]). Structures are taken as the 'objects' of acategory, treated as simples or 'points' by the axioms, and the 'morphisms'between the 'objects' typically preserve the characteristic 'structural prop-erties' of the branch of mathematics in question. Thus, for example, isome-tries preserve metric structure, homeomorphisms preserve topological struc-ture, diffeomorphisms preserve differentiable manifold structure, etc. Cat-egories themselves can be treated as 'objects' in a category, and one canmake sense of morphisms ('functors') preserving structural relations amongthe maps in the original categories. One even makes sense of morphisms('natural transformations') of functors, giving rise to a functor category.(For an overview, see Mac Lane [1986], 386-406. For a categorial recoveryof number theory, see McLarty [1993].)

(iii) Rather than realizing structuralism within an overarching existingmathematical theory, one may pursue a sui generis approach, taking struc-tures to be patterns or universals in their own right. (See e.g., Resnik [1981],Shapiro [1983], [1989], and [forthcoming]; for critical analysis, see Parsons[1990].) Different conceptions under this heading are possible, dependingon the conception of universals. (See Shapiro [forthcoming].)

(iv) A modal-structural ('ms') approach, as in MWON. Here literal quan-tification over structures and mappings among them is eliminated in favorof sentences with modal operators. (Hence the term 'eliminative structural-ism', see Parsons [1990], Shapiro [forthcoming]. And hence the title of thispaper, which I owe to Shapiro.) The framework is a modal second-orderlogic with a restricted (extensional) comprehension scheme. (For details, seeMWON, Ch. 1.) Categorical axioms of logical possibility of various typesof structures replace ordinary existence axioms of MT or CT, and typi-cal mathematical theorems are represented as modal universal conditionalsasserting what would necessarily hold in any structure of the appropriatetype that there might be. It turns out that a great deal of ordinary math-ematics may thus be represented nominalistically, without the language of



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classes at all, even under modality. (See Hellman [1994] and below.) Justhow far this approach can be pushed is a somewhat open question, to bepursued further below. This brings us to the question already broached atthe outset, whether this approach can do justice to 'structuralism' withouta detour through (modal-structurally interpreted) set theory or categorytheory. In the following sections, we will present some evidence in favor ofa positive answer.

Now this is not the place to undertake a systematic comparison of thesealternative approaches. There are, however, two related contrasts of im-mediate concern between (i) through (iii) on the one hand and (iv) on theother that require our attention. The first pertains to the trade-off betweenplatonist ontology and modality. The first three approaches are framed inmodal-free languages but they are entangled well above the neck (naturally)in Plato's beard. Sets, categories, or universals are just taken as part of real-ity, leading to perennial disputation as to the nature of such 'things', how wecan have knowledge of them or refer to them, etc., and (of course) whetheror not such questions are somehow misguided in the first place. Modalstructuralism avoids commitment to such abstracta, at least in its initialstages (in treating, say, the number systems, prior to reconstructing set the-ory itself), and raises the prospect that a (modal) nominalistic frameworkmay suffice to represent the bulk of ordinary mathematics.1 (This dependson treating the second-order variables of the ms language nominalistically,but in ordinary contexts this can be done (see Hellman [1994] and below).)The price of course is taking a logical modality as primitive, raising ques-tions of evidence and epistemic access not unlike those raised by platonistontololgies. This trade-off is a subject of ongoing discussion, and will notbe resolved here. We would point out, however, that assessing the trade-offdepends on a better understanding of the alternatives themselves, includingthe ms approach. In particular, just what modal-existence postulates are re-

1 The phrase 'ordinary mathematics' is not a precise one, but we Intend It more broadlythan do Friedman-Simpson et a/, in the program of reverse mathematics, where explic-itly excluded are 'those branches of mathematics which ... make essential use of theconcepts and methods of abstract set theory', such as 'abstract functional analysis, gen-eral topology, or uncountable algebra'. (Brown and Simpson [1986], p. 123.) We domean to exclude set theory and category theory themselves, but not the three fields Justlisted, nor the theory of non-separable Banach and Hilbert spaces, which by implicationare also excluded by the Friedman-Simpson usage. The latter Is motivated primarilyby the question, 'What portions of ordinary mathematics can be carried out in whichinteresting subsystems of classical analysis (PA2)?' and for this purpose, mathematicalquestions which cannot even be asked (even via suitable coding) in the language of PA2

are sensibly excluded from 'ordinary mathematics'. Since we are under no such con-straint, however, we can afford to be more liberal, counting as ordinary virtually anysubfield short of those devoted to the grand foundational schemes. In our usage, certain'concepts and methods of abstract set theory' can be deployed to some extent withoutcommitment to abstract sets. But this can be spelled out without a precise use of thephrase 'ordinary mathematics'.



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quired to implement structuralism! The more modest they are, the betterthe prospects for the ms approach. The results of our reflections below willbear directly on this. As will emerge, only rather modest modal-existencepostulates are required; for much of mathematics, only countably manyatoms need be postulated (as logically possible); for much more, includ-ing a great many topological structures and manifolds, uncountably manyatoms are needed, but this need not transcend the scope of nominalism.

The second contrast between (i)-(iii) and (iv) concerns the wealth ofmathematical structures incorporated within the respective framework. Inthe cases of (i)-(iii), the extent of richness is literally endless. While anyparticular set theory has its limitations, there are still boundless riches asregards the structures and spaces of ordinary mathematics. (In particular,there are no limits on cardinality or on type.) Category theory, especiallywith its large categories, is prima facie even more generous. And, presum-ably, sui generis universals are, as Quine might say, 'free for the thinkingup'. Not that these frameworks avoid honest toil; nor that they are larce-nous; they merely rely on the bountifulness of reality as they conceive it.

The case of the ms approach is more complex; indeed, we should distin-guish two sub-approaches: (a) first develop a modal-structural interpreta-tion of set theory (or of category theory), and then simply translate the MT(or CT) treatment of structures of interest accordingly; (b) seek a directms interpretation of theory of any such structures, avoiding set-theoreticcommitments to whatever extent possible. From the perspective of onto-logy, it is (b) that is of greater interest. Moreover it confronts the challengeof describing interrelations of different types of structures, something thatboth MT and CT are set up to handle. If approach (b) were to be suc-cessful, structuralism would then stand independently of set theory ratherthan being just a chapter in it, even as interpreted; and it would representa rather remarkable extension of nominalistic methods. It is this approachthat we shall now continue to pursue.

(Traditionally, the problem with nominalism in mathematics has beennot so much that Occam's razor has been dulled by Plato's beard, butrather that it has managed to remove the beard only by severing the headat the neck. Modal structuralism, as it has been extended (in my [1994] andbelow), manages a fairly clean shave while leaving the brain quite intact.)

2. Extending the Reach of Nominalism to Third-orderArithmetic and Third-order Analysis

The plan of this section is as follows. First we shall review the modal-structural frameworks for arithmetic and for real analysis developed inMWON, taking advantage of certain improvements since developed. Theseimprovements consist principally in (1) the combined use (due to Burgess,Hazen, and Lewis [1991], henceforth 'BHL') of plural quantification (Boolos



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[1985]) and mereology to define riominalistic ordered pairing in a generalway (as opposed to adopting a new primitive pairing relation, as was sug-gested in MWON); and (2) the development of predicative foundationsfor arithmetic (in Feferman and Hellman [1995]) which enables modal-structuralism to get started, at least, in a manner compatible with pred-icativist principles. Having reviewed this, we will then indicate how themachinery just referred to under (1) can be used to extend the reach ofnominalism one level beyond each of the core systems of arithmetic andreal analysis described in MWON-essentially how to pass from PA2 to PA3

and from RA2 to RA3. Some of the benefits of these extensions regardingstructuralism will then be explained in the next section.

Beginning with the standard Peano-Dedekind axioms for the naturalnumbers, PA2, involving just successor,', and the second-order statement ofmathematical induction, we treat an arbitrary sentence 5 of first- or second-order arithmetic (in which any function constants have been eliminated bymeans of definitions in terms of') as elliptical for the modal conditional

in which a unary function variable / replaces ' throughout and the super-script X indicates relativization of all quantifiers to the domain X. This isa direct, modal, second-order statement to the effect that 'S holds in anymodel of PA2 there might be'. (Note that use of model-theoretic satisfactionis avoided.) This was called the 'hypothetical component' of the modal-structural interpretation (msi) of arithmetic. In order that this provide afaithful representation of classical arithmetic, it is also necessary to add a'categorical component', a statement that such structures (w-sequences orN-structures) are logically possible:

o 3X3f[APA2]x ('//). (P°ss N)

This is the characteristic modal-existence (mathematical existence) claimunderlying classical arithmetic (or 'classical analysis', logicians' term forPA2). It distnguishes the modal-structural approach from 'deductivism'and from 'if-thenism'. All sentences of the original mathematical language(for PA2) are regarded as truth-determinate regardless of their formal prov-ability or refutability. Furthermore, various arguments show that the trans-lation scheme respects classical truth-values. A key step is the recovery ofDedekind's categoricity proof, that any pair of models of PA2 are isomor-phic. This can be carried out within modal second-order logic, using justthe ordinary second-order (extensional) comprehension scheme (with ordi-nary universal quantifiers in the prefix, not boxed ones) and basic quantifiedmodal logic (although S-5 is the preferred background). Here the reasoningis straightforward mathematical reasoning under the assumption that a pair



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of PA structures is given. Appeal to intensions—relations across possibleworlds, as it were—can be avoided if we assume an 'accumulation principle'to the effect that if it is possible there is an w-sequence with (PA2-definable)property P (which, by quantifier relativization, involves only items internalto the given sequence) and it is possible there is another w-sequence withsuch property Q (internal to its sequence), then the conjunction of theseexistential statements is also possible, i.e., these two sorts of (^sequencesoccur in the same world, so to speak. (Fbr further details, see MWON,Ch. 1; also Hellman [1990].)

Note that talk of 'possible worlds' is heuristic only; the modal operatorsare primitive in the framework and are not required to be given a set-theoretical semantics. Note further that the accumulation principle derivesits plausibility from the combination of two considerations: first, it is onlylogical possibility that is at issue, and second, the mathematical propertieslabelled 'P ' and 'Q' are entirely 'internal' to their respective sequences, asrelativization to the respective domains of any quantifiers they may containinsures. The essential point is that anything internal to a given structurecannot conflict with anything internal to another, so that structures sa-tisfying the respective conditions are logically compossible. Thus, there isno requirement that the structures involved be of the same general type.One could be an ^sequence and another could be an ordered continuum,or whatever. Moreover, the principle can be generalized in the obviousway to cover any finite number of structures. These generalizations areimportant for this approach to structuralism, since, as in set theory orcategory theory, we often wish to speak of relations among a variety ofstructures. Finally, note that the formulas above quantify over structuresby quantifying directly over their domains and distinguished relations orfunctions; it is not necessary to ascend a further level in type as is commonlydone in model theory. From the rest of the formula, it can always be madeclear which relations or functions are defined on which domains.

So far we have used some of the ordinary language of mathematics—thelanguage of 'domains' and 'functions'—to eliminate reference to 'numbers'as special objects. Arithmetic is not about special objects; it is ratherabout a special type of structure. In accordance with part of Dedekind'sconception, it investigates facts which hold of any 'simply infinite system',but we have explicitly used modal operators both to get away from commit-ments to any special instantiation of the structure-type and to achieve anopen-ended generality appropriate to mathematics.2 On this conception,

2 As TaJt [1986] and Parsons [1990] have pointed out, it would be a mistake to attributeto Dedeklnd himself an eliminatlvist structuralist position (cf. MWON, Ch. 1). In thetext, however, we are referring to Dedekind's conception of 'the science of arithmetic' asinvestigating what holds in any simply infinite system. This, surely, is the starting pointof any eliminatlvist approach.



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mathematics investigates a certain category of necessary truths, not con-fined to what happens to exist. But it does not have to postulate a specialrealm of necessary existents in the process. Only one level of abstracts hasbeen invoked, corresponding to the second-order variables. And, it shouldbe noted, the second-order comprehension scheme does prima facie commitus to actual classes and relations of whatever actual first-order objects werecognize among the reiata of the relations (or relation variables) of ourlanguage. Significantly, there is no iteration of collecting, so this is notPlato's full beard, to be sure; but it does seem more than just a six o'clockshadow! So how do we get a clean shave?

Well, it turns out, there are many ways to shave a beard, at least at thestage of second-order arithmetic. One way is that of predicative founda-tions, in the tradition of Poincare, Weyl, and Feferman et ai. The centralidea here is to restrict comprehension axioms to definable classes (and re-lations), where this is spelled out in terms of formulas of mathematicallanguage whose quantifiers range over already defined or specified objects.(For details on various options, including systems of variable type, see writ-ings of Feferman, e.g., [1964], [1968], [1977], [1988].) Typically one beginsby taking the natural numbers for granted and considering, first, those setsof natural numbers definable by arithmetic formulas (with quantifiers onlyover natural numbers)—the first-order sets—and then sets of natural num-bers definable by formulas with quantifiers over numbers and first-ordersets, and so on. (How far this may be iterated is a delicate matter.) Thiscan qualify as nominalistic—relative to the natural numbers—in that onecan eliminate reference to sets and relations in favor of the semantic notionof satisfaction of formulas by natural numbers, or by other nominalisticallyacceptable objects, e.g., predicates themselves. (Cf., e.g., Chihara [1973],Burgess [1983].)

The problem with this as a nominalization program, however, is thatthe natural-number structure has been taken as given. And this has ap-peared unavoidable, for there is prima facie reliance on impredicative classexistence principles in the classical constructions of the natural-numberstructure, e.g., the Dedekind-Frege-Russell definition as (essentially) theintersection of all inductive classes containing 1 (or 0). Such principlesare standardly used also to prove the existence of an isomorphism betweenany structures satisfying the PA2 axioms; and it is well known that thesecond-order statement of induction is necessary for this result. Contraryto these appearances, however, the natural-number structure can itself beconstructed predicatively, beginning with the notion of finite set governedby axioms that are intuitively evident or of a stipulative character, as car-ried out by Feferman and Hellman [1995]. Of particular interest here arethe facts that mathematical induction is itself derivable from within an ele-



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mentary theory of finite sets and classes ('EFSC'),3 and that Dedekind'scategoricity proof ('unidty of the natural number structure') is also recov-erable. (Cf. Feferman-Hellman [1995].)

This still leaves us with the non-nominalist notion of 'finite set' governedby the EFSC axioms. But even this vestige of platonist commitment (5o'clock shadow?) can be eliminated. One can first postulate the logicalpossibility of an infinitude of atoms (atomic individuals, governed by theaxioms of atomic mereology; cf. Goodman [1977], also MWON, Ch. 1), andthen interpret 'finite set' as 'finite sum (or whole, or fusion) of atoms'. Toexpress 'infinitude of atoms' one can use the device of plural quantificationand postulate:

'There are (possibly) some individuals one of which B an atom and eachone of which fused with a unique atom not overlapping that individualis also one of them.' (Ax oo)

With this postulate, one has the essentials of a mereological model of theEFSC axioms: first-order variables can be taken to range over arbitrary in-dividuals (atoms and fusions of atoms); finite set variables range over finitefusions of atoms; class variables range over arbitrary fusions of atoms. Ifa null individual is admitted (as a convenience), the axioms are satisfiedas they stand; otherwise the comprehension axioms can be complicatedslightly to avoid the null individual. (EFSC also takes a pairing function asprimitive, governed by two axioms: P-I, the standard identity condition,(x,y) = (u, v) iff x = u and y = v, and P-II, existence of an urelement un-der pairing. It turns out that both-these are satisfied on, say, the Burgessconstruction of nominalistic pairing in BHL [1990]. So we can invoke thisin interpreting pairing in EFSC.) Now, within such a model of EFSC thereis a mereological model of the PA2 axioms. (This follows from the con-

3 The system EFSC Is formulated In a three-sorted language with individual variables,variables for finite sets of individuals, and variables for classes of individuals. Formulaswith no bound class variables are called 'WS' formulas (for 'weak second-order'). A pair-ing operation-symbol is primitive as is € relating individuals to finite sets and classes.The logic is classical (with equality in the first sort). The axioms of EFSC are, in words,as follows:(WS-CA) Weak second-order comprehension: existence of classes as extensions of

WS-formulas;(Sep) Separation for finite seta: existence of a finite set as the intersection of any

given finite set and the extension of a WS formula;(Empty) Existence of the empty finite set;(Adj uction) Existence of a finite set obtained by adjoining any single lndividal to a given

finite set;(Pairing I) 'Pairs are distinct Just in a case either first or second members are';(Pairing II) Existence of ureletnents under pairing.

The system EFSC* is obtained from EFSC by adding the axiom,(Card) 'Any finite set is Dedekind-finite.'

In Feferman-Hellman [1995], the existence of structures satisfying mathematical in-duction is derived in EFSC*; however, as Peter Aczel has pointed out, this can alreadybe proved in EFSC.



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struction of an N-structure in EFSC.) Moreover, as the class variables aretaken to range over arbitrary individuals (fusions of atoms), we even havea full second-order PA2 model in the classical sense, in which arbitrarysets of numbers correspond to arbitrary fusions of the individuals servingas numbers. The predicativist may stop short of this, confining oneself to'definable fusions' of atoms in specifying the range of the class variables.(Cf. Hellman [1994], sec. 3.) But the essential point here is that both pred-icativist and full second-order arithmetic are interpreted nominalbtically.

Prom here, one could continue on with predicativist analysis, construct-ing countable analogues of the classical continuum, made up of 'definable'or 'specifiable' reals, which support rather rich portions of functional anal-ysis and related subjects. (Cf., e.g., Feferman [1988].) As Fefennan haspointed out, within various systems of predicative analysis, one can evenprove the unicity of the real number structure, as seen from within that sys-tem: impredicativity is avoided because one requires, not the full classicalprinciple of Continuity (least upper bound axiom), but only the sequen-tial form, 'Every non-empty bounded sequence of reals has a least upperbound'. (For details, see Hellman [1994], sec. 2.)

Continuing along the predicativist route, one can introduce symbolismfor reasoning about (specifiable) classes (and relations and functions) ofreals, classes (etc.) of classes of reals, and so on through the finite types.(See, e.g., Feferman's system W [1988] and [1992].) At each level, one isconsidering not the full classical totalities, of ever higher uncountable cardi-nality, but subtotalities of objects predicatively specifiable in mathematicallanguage.4 Thus, the ranges of the quantifiers at each level are really count-able, although from within the predicativist system they may be describedas 'uncountable'. (The predicativist can carry out the reasoning of Can-tor's diagonal argument, but, implicitly, only specifiable enumerations areconsidered.)

To what extent can predicativism carry out a structuralist program formathematics? This is a large question which cannot be fully answered here.But the following points towards an answer may be offered. First, one mustbe more precise about what it means to 'carry out a structuralist program'.Presumably this includes these things: (i) characterizing the types of struc-tures or spaces that arise in the various branches of mathematics (or at least'ordinary mathematics' as we have used that term above); (ii) describingthe main types of relationships among these structures, including the var-ious morphisms within and among them (isomorphisms, homomorphisms,embeddings of various sorts, etc.); (iii) recovering the important theoremsconcerning the various structures and relations among them, including ex-istence theorems.

* For ways of making this precise In connection with unramified systems such as W,see my [1994], n. 2.



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Judged by these standards, predicativism gets mixed reviews. It doessurprisingly well, for example, in recovering theories of various types of met-ric spaces central to scientifically applicable mathematics. Although theconcept of Lebesgue outer measure is not predicatively available, theoriesof measurable sets and measurable functions can be developed (cf. Fefer-man [1977], §3.2.5), and then one can obtain the Lp spaces and carry out astructuralist treatment of Banach and Hilbert spaces. On the other hand,clearly there are important limitations: 1) Various objects of importancein the classical (set-theoretic) treatments are simply not available, e.g.,outer measures, as just indicated, and general descriptions of major typesof spaces are prima facie impredicative, e.g., topological spaces with fami-lies of open sets closed under arbitrary unions. 2) Proofs of key theorems,even if predicatively statable, may require impredicative constructions es-sentially. A known example is Friedman's finite form of Kruskal's theoremon embeddability of finite trees. (See e.g., Smorynksi [1982].) Even if thisexample does not pertain to scientifically applicable mathematics, it surelypertains to significant mathematical structures. 3) Even in the cases inwhich predicative proofs of key theorems are possible, e.g., the unicity ofthe real-number structure, in reality we know—as does the predicativist—that the structures to which the result pertains are countable, hence onlysmall parts of the structures classically conceived. Even if the predica-tivist makes no direct sense of the latter phrase, one can pass to moreencompassing, predicatively graspable totalities, essentially by enriching agiven language with predicatively intelligible semantic machinery for defin-ing new objects, e.g., real numbers. (Cf. Hellman [1994], §2.) If one isa skeptic about uncountable totalities generally, then presumably one iswilling to pay the price of this language-relativity of much of mathematics.If, however, one follows the classicist in taking seriously the absolutenessof uncountability—e.g., if one treats totalities such as 'all sets of naturalnumbers' or 'all fusions of countably many atoms' as having a definite andmaximal sense—then one will regard the predicativist substitutes as fallingfar short of the genuine articles.

This much should be clear: If the objection to the uncountable is moti-vated by nominalist concerns—the desire to avoid commitment to classesor universals, etc.—then it is misplaced. A fusion of atoms is just as'concrete'—just as much not a class or a universal, etc.—as the atoms them-selves. In the language of types, both are of type 0. It does not matterhow many there are. If we are given countably infinitely many atoms—by definition pairwise discrete—then we may speak of arbitrary fusions ofthem without nominalistic qualms. Then to go on to say how many of suchfusions there are requires further reasoning, to be sure; but in fact thereis no problem in carrying out Cantor's diagonal argument nominalisticallyto convince oneself that there are uncountably many. And if in the course



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of this reasoning, one meant to consider 'any possible enumeration'—notmerely any that could be specified in some privileged symbolism as thepredicativist intends—then the conclusion has its absolute force.

This leads us to relax the 'definitionist' stance of predicativism in pursuitof a nominalist structuralism. In our nominalist E-comprehension scheme,

3i*(x) - 3uVy[j/ o « « 3z(*(z) & z o y)} (CE)

(in which o is 'overlaps' or 'contains a common part with'), we allow thepredicate <I> (lacking free u) to contain quantifiers over arbitrary individuals,whether or not specifiable by any particular symbolic means. An immedi-ate consequence is that, once we have postulated (Ax oo)—guaranteeingan w-sequence of atoms (Poss N)—we already have embedded within sucha sequence enough subsequences to serve as arbitrary real numbers. Stan-dard arithmetization procedures can be used to introduce negative inte-gers, rationals, and then reals (either as Cauchy sequences of rationals oras Dedekind cuts). (By the device of numerical pairing, that is pairing viathe atoms of the postulated ^sequence, one can remain within that struc-ture; reals are then just certain fusions of atoms.) The following importantfacts should be noted:

(1) The full classical Continuity principle (lub principle for arbitrarynonempty bounded sets of reals) is derivable along bgicist lines (using CE),without exceeding the bounds of nominalism. (The usual set-theoreticalarguments are avialable making use of plural quantifiers to get the effect ofquantification over sets and functions of reals.)

(2) The categoricity of real analysis (RA) is also derivable within thisnominalist framework, in the sense that any two concrete R-structures (i.e.,with reals built up from concrete w-sequences in logicist fashion as just al-luded to, together with the usual ordering < on reals) are isomorphic. (Theproof of this requires even less than (1), viz. Sequential Completeness ratherthan the full lub principle suffices. For a visualizable nominalistic construc-tion, see Hellman [1994].) Furthermore, since all fusions of atoms count asindividuals, regardless of specifiability by formulas, this categoricity proofhas the absolute significance of the standard set-theoretical one.

Thus, (Poss N)—hence (Ax oo)—suffices for a nominalist structuralisttreatment of full classical analysis (PA2). In particular, the criteria (i)-(iii)above are met with respect to PA2-structures. But, as the alert reader mayhave noticed, we have enough machinery at our disposal to ascend one morelevel, to third-order number theory (PA3). Plural quantifiers achieve theeffect of quantification over sets of reals, as just described; and the deviceof BHL pairing reduces polyadic quantification at this level (over relationsof reals) to monadic. Still we have only had to hypothesize a countableinfinity of atoms.



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There may be the concern that plural quantifiers (at a given level, saypluralities of reals) do not really get around classes (of reals); that a sentencesuch as, 'Any reals that are all less than or equal to some real are all lessthan or equal to a least such', really concerns classes of reals as valuesof a hidden variable. (No first-order conditions determine the same classof models.) I agree with Boolos [1985] and others, however, that we dohave an independent grasp of plural quantifiers and their accompanyingconstructions, and that we can use them to formulate many truths at anygiven level that would have to be regarded as false on an ontology thatrepudiates classes of objects at that level. (The case based on this andrelated points has been made very effectively by David Lewis in his [1991],§3.2.) The school teacher who says, 'I've got some boys in my class whocongregate only with each other', should not be ascribed on that basis acommitment to classes other than school classes. The EPA official who says,'Some cars are tied with one another in being the most polluting vehicles onthe road', need not be committed to classes of vehicles other than the usualpredicative ones (two-door sedan, station wagon, etc.). And the nominalistwho entertains just some atoms and their fusions can even go on to saythings like, 'Some of those fusions can be matched up in a one-one mannerwith the atoms whereas not all of them can', without implying anythingabout any objects of higher type than the fusions themselves. (In virtue ofBHL pairing, even the talk of one-one correspondences is innocent.)

Thus the strength of full, classical third-order number theory—equiva-lently, second-order real analysis—is attained within a nominalist modal-structural system without postulating more than a countable infinity ofatoms. This is already quite a rich framework for carrying out structural-ism. But if we are prepared to entertain at least the logical possibility of acontinuum of atoms, we can attain one full level more, that of fourth-ordernumber theory, equivalently third-order real analysis. Suppose we postulatethe possibility of a complete, separable ordered continuum ('R-structure',for short) of atoms, which we may write

o3X3/[/vR/!2]x(< / / ) , (Poss R)

in which RA2 denotes the axioms for such a structure (with first-order quan-tifiers stipulated to range over atoms), including the second-order statementof Continuity, and where a relation variable / replaces the ordering rela-tion constant < throughout; then our second-order variables already rangeover arbitrary fusions of these, at the level of sets of reals. Functions andrelations of reals are reducible to sets via pairing. Quantifying plurally oversets of reals then gives the effect of quantifying (singularly) over sets of setsof reals and, via BHL pairing, over functions and relations of sets of reals(hence also of functions and relations of reals). This is mathematically atthe level of RA3 or PAA, a very rich framework indeed. (Note that, since



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fusions of (atoms serving as) reals are entertained as bona Bde objects, it ispluralities of them that we plurally quantify over to achieve the next level.Even if plural quantification is already invoked lower down, to introducepairs, triples, etc. of reals so as to reduce relations of reals to sets, we getthe effect of plurally quantifying over relations of reals when we plurallyquantify over sets of reals. We are not 'plurally quantifying over pluralities'except in this innocent sense.)

In order to characterize R-structures, it is necessary that we use thesecond-order statement of the Continuity principle (lub axiom); it is one ofthe defining conditions built into the modal-existence postulate (Poss R).Does this mean that we have given up the advantages of a logicist-stylederivation of Continuity from more elementary, general principles? Well,yes—but also no! Yes, in that the principle (or a geometric equivalent)is essential in asserting the possibility of an R-structure whose first-orderobjects are atoms. But no, in that we have already derived Continuity inlogicfet fashion (but nominalistically) above as it governs R-structures builtfrom fusions of atoms of an N-structure. If reals are taken as constructedobjects (e.g., Dedekind sections, etc.), then we already have (Poss R) basedon (Poss N), as already described. (This point was not made in MWON, for,not utilizing the resources of plural quantifiers, we did not have the meansto state Continuity in full generality, nominalistically, given just (Poss N).A major advantage of the present approach is that, now, we do have themeans to speak of arbitrary pluralities of reals, given just (Poss N).) Wegive none of this up when we add the postulate of an atomic R-structure.But of course we are adding something substantial. In terms of the familiaraphorism, there are two cakes: we are having one while eating the other.

Now the worry arises that in entertaining anything so idealized andremote from experience as spatio-temporal or geometric points (typicallythought of as realizing the RA axioms) we have exceeded the bounds of nom-inalism. Surely a case can be made that 'points of space-time'—especially'unoccupied points'—are in some sense 'abstract', even if invoked by stan-dard formulations of physical theory. Field [1980], who based his nomi-nalization program on the acceptability of space-time points and regions,sought to answer such objections, in part by appealing to substantivalistinterpretations of space-time physics which do seem to accord a kind ofcausal role to (even unoccupied) space-time (e.g., as a field source withclearly physical effects). And certainly this neo-Newtonian perspective isnot without its adherents. But it is important to realize that the modal-structural axiom, (Poss R), does not ultimately depend on a substantivalistview of space-time. Fbr, just as the logical modality frees us from having toposit actual countable infinities, so it frees us from any particular view ofactual space, time, or space-time. It suffices, for example, that Newtonianswho posited a continuous luminiferous medium were entertaining a logically



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coherent possibility. Similarly one can coherently imagine a perfect fluidfilling, say, a bounded open region and satisfying whatever causal condi-tions one likes. All that matters is that it be conceived as made up of partssatisfying the RA2 axioms; it need not involve 'unoccupied space'; it neednot be useful to any explanatory purpose; it need not be (period). Fur-thermore, supposing that such possibilities are 'world-independent'—thatthey have the absolute status generally accorded logical possibilities—freesmathematics based on (Poss R) of any contingency. (In the modal systemS-5, appropriate to logical possibility, we have Nee Poss R once we have PossR.) There is no 'lucky accident' involved in the truth of modal-structuralistmathematics.

Since the comparison with Field's program has come up, it is appropri-ate to emphasize that, for Field also, there is no 'lucky accident' involved inthe truth of mathematics, for (except in a vacuous sense) Field's programis instrumentalist, not recognizing mathematical truth at all, except in vac-uously assigning 'false' to all existential statements and 'true' to anythingequivalent to the negation of one, since 'there are no numbers'! This isas great a contrast with modal-structuralism—nominalist or not—as therecould be.

Yet perhaps the contrast is at bottom illusory. For, in seeking to recovernominalistically various arguments that certain mathematical-physical the-ories are semantically conservative with respect to their nominalized coun-terparts, Field [1989] [1992] introduced logical modality as a means of by-passing reference to models. Along the way he seems committed to affirmingsuch things we would write as,

oAPA2, (FC)

that the conjunction of the second-order Peano-Dedekind axioms is logicallypossible. ('FC is for 'Field's commitment'.) Now this is not quite the sameas our (Poss N), which involves domain and function variables, but (PossN) follows in second-order logic from (FC). Mathematically speaking, theyare not essentially different. (Philosophically they are. As explained inMWON, Ch. 1, problems arise in interpreting the function constant of (FC)under the modal operator, problems that (Poss N) avoids. And explicitreference to domains is natural in a structuralist interpretation.) But theneverything needed for a modal-structuralist treatment of mathematics inPA2—or even PA3, if plural quantifiers are added—is available. So Field'ssystem is not really instrumentalist after all. It embraces the core of thems interpretation but then does not go on to utilize it. In this sense, muchgenuine mathematics is still present.5

B Concerning Field's commitment (FC), let 'Ax' stand for the conjunction of axioms



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3. Realizing StructuralismWithout attempting anything like a survey of mathematical structures, letus indicate by salient examples how the above described frameworks sufficefor a great deal of the structuralist enterprise. Let us refer to those frame-works as PA3 (or RA2) and PA4 (or RA3), understanding of course that itis the nominalistic modal-structural version of these systems that is meant,that is, the relevant modal-existence postulate, e.g., (Ax 00), with the back-ground of S-5 modal logic, mereology with the comprehension scheme (CE)for modal-free formulas, plural quantifiers, and the assumptions needed forBHL pairing (as in the Appendix of Lewis [1991]). (The second-order logicalnotation of the modal-existence postulates may be retained, understandingthe monadic second-order variables to range over arbitrary fusions of atomsand polyadic second-order variables to range over fusions of n-tuples of in-dividuals based, say, on b- pair ing of BHL.) It should also be understoodthat, in treating many abstract structures, what we really require of theseframeworks is not the full structure of an N-structure or an R-structure,but merely a domain with sufficiently many atoms, e.g., a countable infin-ity or continuum many. Then one needs the specific functions, relations, orother items characteristic of the type of abstract structure in question, e.g.,a metric, a topology, a chart-system, etc. Thus when we appeal to PA2 (orPA3), say, we may only be appealing to the modal-existence of a countableinfinity of atoms; and when we appeal to RA2 (or RA3), we may only beappealing to the modal-existence of continuum-many atoms. Alternatively,we may only wish to appeal to the modal-existence of uncountably manyatoms, leaving open the cardinality. It should be clear how to formulatethis using the available machinery, mereology, plural quantifiers, and BHLpairing. We need merely translate the statement,

is a fusion of infinitely many atoms & a proper part Y of X is

of a finitely axiomatized mathematical theory T (such as PA3), and let 5 be a nomlnal-istlcally formulated statement (of applied mathematics); then Field's modal formulationof the conservativeness of T is

If oS, then o (Ax & S), (Conserv T)where Field's © operator is said to mean 'it is not logically false that'. (Cf. Field [1992],p. 112.) Initially, Field focuses on first-order theories (for which (Conserv T) may bea schema with finite conjunctions of axioms), but later he considers some second-ordercases as 'not completely without appeal'. Moreover, since the 'complete logic of Good-manian sums' is invoked (in Field [1980]) in connection with representation theoremsfor R*—i.e., even arbitrary fusions of space-time points are recognized—there shouldcertainly be no objection to (Conserv PA2). (Indeed, as we have seen, Ax in this case isnominalistically Interpretable without anything so strong as space-time substanttvalifim.Field's reservations in connection with second-order principles seem to have primarily todo with transfinite set theory and not with arithmetic or analysis. Cf. his [1992], p. 119.)But of course there are some S such that Field accepts oS, and then oAx follows.

It may well be that Field's Informal understanding of the logical modality differs insome respects from that expressed in MWON, but his core modal logic, like that ofMWON, is a version of S-5, and it seems clear that a great deal of modal mathematics(of PA2 and even RA2) can be carried out within Field's system.



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a fusion of countably many atoms (which we can say directly by sayingY is in one-one correspondence with any infinite part of Y) & X is notin one-one correspondence with V].

If we wished to specify a fusion of Ni atoms, we could simply add to thisthat X is in one-one correspondence with any infinite part which is not inone-one correspondence with Y. Similarly we could specify a fusion of N2atoms, and so on. (Writing such things out in primitive notation with allthe plural quantifiers needed for BHL pairing would not be pleasant, butthat is why we like quotes. Fbr more on the expressibility of cardinality,see Shapiro [1991].)

Let us begin with the important example of metric spaces. Abstractlyconsidered, these are pairs consisting of a domain X and a function d (pos-itive definite metric) from X x X into R satisfying the familiar first-orderconditions: d{x,x) = 0, x ^ y —* d(x,y) > 0, d(x, y) = d(y,x), andd(x,y) + d(y,z) > d(x,z) (triangle inequality). Bijective mappings <j> be-tween two such spaces, (X, d) and (X',d')t preserving metrical relations arecalled isometries. (That is, d(x, y) = d'(<f>(x),<j>(y).) Even if we require thatthe domains be uncountable, all this is describable at the level of PA : thedomains are at the level of sets of reals, metrics are at the level of sets ofordered triples of reals, hence sets of reals via pairing, and isometries are atthe level of sets of ordered pairs of reals, hence sets of reals. Thus we canquantify over structures of this type and isometries (and similar relations)among them without positing more than a countable infinity of atoms.

Of course, many metric spaces of importance carry additional structure,often embedded in the elements of the spaces themselves. Fbr example,Banach spaces are normed linear spaces whose elements are often real- orcomplex-valued functions; they become metric spaces under the definitiond(f,g) = | | / — g\\. Hilbert spaces have the additional structure of an in-ner product, ( , ), which gives rise to a norm via ||/||2 = ( / , / ) . Suchspaces are called separable if they include a countable dense set of vectors(where density means that any / in the space can be approximated ar-bitrarily closely in the metric by elements of the countable subset). Thevectors of a separable Hilbert space are codable as real numbers, and innerproducts are then at the level of sets of reals (via pairing). Linear oper-ators are also at this level, as are subspaces. The latter, however, can becoded as reals since a subspace is spanned by countably many vectors. Thesame is true for a great many operators, e.g., the continuous (= bounded)ones. Norm-preserving maps, unitary transformations, are also at the levelof sets of reals. Thus, the metrical structure of separable Hilbert spacesis describable in the PAS framework, again not exceeding the postulationof countably many atoms.6 Significantly, the classical demonstration that

6 Indeed, If the vectors of the dense set are codable as natural numbers, as they are



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the axioms for separable Hilbert spaces (of infinite dimension) are cate-gorical carries over intact. (The standard examples Z2 and L2 are avail-able.) Indeed, even measure-theoretic structure can be captured at thislevel. Note, finally, that—in virtue of the unrestricted second-order (exten-sional) comprehension principles of our background logic—theorems of thestandard classical theories of these structures translate into theorems of themodal-structuralist framework, as in the cases of arithmetic and analysis.(Cf. MWON, Ch. 1.) This preservation of theorems holds regarding all thestructures we shall consider, so we need not keep repeating the point.

For some theorems involving multiple structures, instances of accumula-tion principles, described above, may be used. It should be mentioned, how-ever, that in many cases, appeal to such principles can be bypassed and onecan reproduce constructions and theorems regarding multiple structures—indeed even infinite classes of structures—by considering certain parts of asingle possible universe of infinitely many atoms, each such part endowedwith relevant mathematical structure. The example of product spaces intopology is a good illustration and will be described briefly below. (Whetherthe class of structures can be uncountably infinite depends on the case.)

Let us consider now some structures from measure theory, which hasposed a challenge to various constructive programs. (We follow Halmos[1974].) Central are measure spaces, triples (X,S,/J.) where X is a domainof points (e.g., real numbers), S is a class of (measurable) subsets of X (a(T-ring) whose union is X, and /1 is an extended real-valued, non-negative,countably additive set function on S assigning 0 to the empty set. Now, if5 is the <r-ring generated by a collection of cardinality of R (e.g., 5 is thecollection of Borel sets of reals, generated by the bounded left semi-closedintervals), it has cardinality of R also. Then the members of S can be codedas reals so that /J. is of the type of a function from R to R, at the level ofPA3, and so within the framework of ms arithmetic.

Beyond this, however, one quickly encounters measure-theoretic struc-tures that require resources beyond PA3. Let X be the real line, R1, S theclass of Borel sets, and \JL Lebesgue measure on S. Let S be the result ofadding to 5 all sets of the form EL) N where E € 5 and N is a subsetof a member of S of measure 0, and let ji be the completion of \x on 3(ji(E A N) = n(E), E and N as just described, where A is symmetricdifference), ji is the complete Lebesgue measure, and the sets of S arethe Lebesgue measurable sets. Now, since there are uncountable (Borel)sets of (/i) measure 0 (of cardinality of the continuum, in fact, by Cantor's

in standard examples such as the V spaces, then the metrical structure of the separableHilbert space is decribable in PA2, and indeed much of the theory can be developed invery weak subsystems. (See Brown and Simpson [1986].) However, the completion ofthe countable dense substructure does not formally exist in PA2; a general structuralisttreatment naturally will distinguish the uncountable completions from their countablecodes, and this requires ascent past PA2.



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middle third construction), and since E G S, n(E) = 0, & F C E togetherimply F G 5, the cardinality of § must be that of the power set of K (byCantor's cardinality theorem). (Cf. Halmos [1974], p. 65 (5), (6).) Thus5 and p. are essentially at the level of classes of sets of reals, i.e., at RA3,one step beyond PA3. But RA3 suffices; this structure is describable within(nominalistic) ms analysis, and its modal-existence assured.

Similar remarks apply to the theory of outer measure. The rele-vant structures are extensions of a measure space, (X,S,/J), of the form(X,H(S),fi*), where H(S), the hereditary a-ring generated by S, is thesmallest <7-ring containing the sets of S and closed with respect to (arbi-trary) subsets of such sets, and n* is the outer measure defined on H(S)by

Here again both the class of (outer) measurable sets and the (outer) measureare at the level of classes of sets of reals, hence at RA3 (PA4). Onceagain, the resources of this framework are needed but suffice for this typeof structures.

Turning now briefly to topological spaces, the story is similar. In gen-eral, these are structures of the form (X,O), where A" is a set and O is aclass of open subsets of X such that O contains the empty set and X andis closed under finite intersections and arbitrary unions. This leaves opena vast array of possibilities. At one extreme, the trivial topology containsjust the empty set and X, and at the other extreme the discrete topologycontains every subset of X. In between lie the most familiar topologies ofthe real line, the plane, etc. (the usual topologies of Rn), which are separa-ble, i.e., have a countable base (e.g., the open n-spheres of rational radii).This means that for every point x in an open set U there is a basic openset B such that x G B C U. Thus, every open set in such a topology canbe represented as a countable union of basic open sets. In the case of R",this means that every open set can be represented by a real number, sothat O itseU" can be represented as a set of reals. Thus, such topologicalspaces can be described in PA3, modal-structural arithmetic. (Similarly forthe various spaces encountered in point-set topology.) However, this repre-sentation relies on metrical information which is generally not available. Along story here can be shortened considerably by noting that the discretetopology (on X) is a worst case, concerning cardinality (which is the chiefguide in determining type level). Thus if X has cardinality of the conti-nuum, any topology on X has cardinality no greater than the power set ofthe continuum, i.e., O is at or below the level of a class of sets of reals, atRA3. And morphisms between such spaces (especially homeomorphisms)are generally capturable at RA2. Thus, a very rich variety of topologicalspaces is certainly describable within the nominalist ms framework. Evenif higher set theory (even Morse-Kelley) is needed for a completely general



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theory of topological spaces (free of any cardinality restrictions whatever),clearly a great deal of the subject can be developed on the basis of our twomodal-existence postulates above, that is without officially countenancingsets or classes at all. (Note here that the predicativist alternative does notextend nearly so far, as the very notion of closure of a topology under arbi-trary unions is not available. This serves further to highlight the contrastbetween nominalism and even liberal varieties of constructivism.)

Let us end this cursory tour by mentioning three examples of topologicalspaces of some complexity, as well as importance, that can be treated withinour framework. One example is that of n-dimensional manifolds, of greatimportance in space-time physics. As was already described in MWON,Ch. 3, the general theory of such structures does exceed the reach of RA2,but only by one level. The effect of adding plural quantifiers is that wenow have RA3 at our disposal, and this does suffice to capture maximalsystems of charts, needed to describe manifold structure. It may seemsomewhat remarkable that, for example, so abstract and extensive a workas O'Neill's Semi-Riemannian Geometry [1983] can be translated virtuallyentirely without loss into a nominalistic framework, but that does seem tobe the case.

Topological spaces often arise from set-theoretic constructions out ofclasses of functions, and one might expect that set theory is inevitablyencountered. Our two final examples serve as an antidote. Consider first theconstruction of product spaces of a given family .Fof topological spaces. (Wefollow Kelley [1955].) Suppose that T is countably infinite and that each ofthe spaces, (Xi,Oi) (i G N) has a domain Xi at the level of a set of reals.The domain of the product space Yli^) consists of the Cartesian product]~[{Aj} of the Xi together with the smallest topology for which inverses ofprojections onto open sets in the coordinate spaces form a subbase. Thatis, the domain consists of all functions / from N to the union of the Xt suchthat /(») G X{, i.e., Y[{Xi} = {/ : f(i) G Xi}. Since each / is (codableas) a real, this Cartesian product is at the level of a set of reals, i.e., atRA2. The product topology on this domain is motivated by the requirementthat the projections Pi onto the factor spaces be continuous, i.e., that forU open in Oi, P^~1[U] be open. The sets of this form are stipulated to be asubbase for the product topology, i.e., finite intersections of such sets forma base. These are of the form V = {/ : f{i) G Ut for t e F}, F a finiteindex set (e.g., a subset of N), Ui an open set in Oj. Such V are at the levelof sets of reals, and so the standard set-theoretic construction of the leastcollection of these closed under arbitrary unions is available in RA3. Notethat for this construction, it is not really necessary to recognize the familyT as an object; it suffices if one is given the collection (or whole) of thedomains Xi together with the assumption that for each of these there is acollection (plurality) Oi of open sets of Xi\ and this can be said in RA3 on



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our assumption that the Xi are at the level of sets of reals. Note furtherthat this construction in RA3 depends on the countability of the family ofgiven spaces. If it is uncountable, functions in the Cartesian product willbe at the level of sets of reals, not reals, and the product topology itself willbe a collection at RA4. Obviously, fully general topology transcends RA3.The same limitation does not arise, however, for products of, say, metricspaces, for which the structure is capturable at the level of relations on thepoints.

Finally, consider the notion of a sheaf, which arises as follows. One beginswith holomorphic (complex differentiable) functions / , g, etc. on neighbor-hoods U, V, etc., of a point c in the complex plane. / and g are said to begerm equivalent at c just in case / and g agree on some open neighborhoodW of c such that W G U C\ V. The equivalence class [f]c of functions underthis relation is called the germ of f at c. One then forms the class Ac ofgerms at c and then introduces A=d'\JceCAc, called the sheaf of germs ofholomorphic functions on C. This becomes a topological space upon takingas basic open neighborhoods of \f]c the class of all germs of / at points ofthe domain U of / , that is {[/]c : c G U}. (Neighboring germs come fromthe same holomorphic function.) One then considers morphisms such as thenatural projection p of A onto C sending each germ [f]c to c, which is a localhomeomorphism. Also one has a continuous function F : A —* C such thatF([f]c) — f(c) which can be used to represent all holomorphic functionsby means of cross-sections. (See Mac Lane [1986], pp. 352 f.) Now it turnsout that the holomorphic functions can be coded as reals and, moreover,that germs can also be so represented via power series in z — c convergentin some open circle about c. Thus the sheaf A becomes identifiable as aset of reals, i.e., at RA2, as do the basic open neighborhoods. The topol-ogy O for this space is itself then at the next level, at our familiar RA3.Thus, even this somewhat elaborate and abstract set-theoretic constructionis within the reach of the ms framework. Finally, it should be noted, allthe diagrams of category theory involving these structures and the variousarrows between them can be described as well, as they represent relationsamong finite tuples of structures (illustrating propositions which may, ofcourse, involve universally or existentially quantified variables ranging overstructures or morphisms).

It should be clear that it is not being claimed that set theory is 'neverneeded1, whatever that might mean specifically, or that mathematics 'oughtto' restrict itself to what can be nominalistically described. Rather the pointhas been simply to illustrate the far-reaching scope of the PA3 and RA3

frameworks in the interests of class-free structuralism, to give some ideaof how rich a structuralism one may actually have without yet embracinganything so strong as general model theory or general category theory. All



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these frameworks have their points and their places, and our task has beento understand better just what these are.

There is a sort of corollary worth noting, however, regarding scientific in-dispensability arguments. As Feferman has already emphasized, in connec-tion with predicativist mathematics, it is remarkable how much of scientifi-cally applicable mathematics can be captured within predicativist systems,and this tends to undercut Quinean arguments for set theory based on in-dispensability for scientific applications. (See, e.g., Feferman [1992].) Thisholds a fortiori for nominalist systems as above, for these reach much fur-ther than predicativist systems, as we have already indicated. While someimpredicative constructions do arise at the outer limits of applicable mathe-matics, it would be a real challenge to find anything in the sciences requiringmathematical power beyond the RA3 framework. Thus, indispensability ar-guments should be seen in a new light, not as justifying set theory per se,but rather as helping (to some degree) to justify key mathematical existenceassumptions such as (modal nominalistic) axioms of infinity, including notjust (Ax oo), or (Poss N), and (Poss R), but also the unrestricted compre-hension scheme (CE) and related principles, such as the full comprehensionprinciple of second-order logic. (Cf. Hellman [forthcoming].) Surprisinglyperhaps, if classes are 'genuinely needed', that is probably not because ofscientific applications but rather because of needs from within mathematicsproper, for instance, because they allow the greatest freedom and ease ofconstruction of anything yet devised. But whether and to what extent wehave or can have evidence for the truth or possibility of models of powerfulset theoretic axioms remains unresolved.

Acknowledgement: I am grateful to Stewart Shapiro for helpful com-ments on an earlier draft. I am also grateful to the National Science Foun-dation (U. S. A.) for support, through Award No. SBER9&-10667, of workon which this material is based.



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[1988]: 'Weyl vindicated: Das Kontinuum 70 Years Later', in Tkmi eprospettive della, logica e della fHosofia dell a. scienza contemporanee. Bologna:CLUEB, pp. 59-93.

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ABSTRACT. Recent technical developments in the logic of nominalism make itpossible to improve and extend significantly the approach to mathematics de-veloped in Mathematics without Numbers. After reviewing the intuitive ideasbehind structuralism in general, the modal-structuralist approach as potentiallyclass-free is contrasted broadly with other leading approaches. The machineryof nominalistic ordered pairing (Burgess-Hazen-Lewis) and plural quantification(Boolos) can then be utilized to extend the core systems of modal-structuralarithmetic and analysis respectively to full, classical, polyadic third- and fourth-order number theory. The mathenatics of many structures of central importancein functional analysis, measure theory, and topology can be recovered withinessentially these frameworks.



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